Equivalent definitions of fusion category arising from separability

TL;DR

This paper establishes multiple equivalent characterizations of fusion categories through separability conditions of tensor functors in semisimple multiring categories with duals, with applications to weak Hopf algebras and module categories.

Contribution

It provides new equivalent definitions of fusion categories based on separability properties of tensor functors, connecting algebraic and categorical perspectives.

Findings

Unit object is simple iff tensor functors by non-zero algebras are separable

Describes connectness of weak Hopf algebras via separability

Transfers simplicity between unit objects and module categories

Abstract

For a semisimple multiring category with left duals, we prove that the unit object is simple if and only if the tensor functors by any non-zero algebra are separable (resp. faithful, resp. Maschke, resp. dual Maschke, resp. conservative). This induces a list of equivalent definitions of fusion category. As an application, we describe the connectness of a class of weak Hopf algebras by the separability of tensor functors. We also consider applications to transfer of simplicity between the unit objects, semisimple indecomposable module category and Grothendieck ring.